Normal subgroups of nonstandard symmetric and alternating groups

John Allsup and Richard Kaye

School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

Published in: Archive for Mathematical Logic, Springer Berlin / Heidelberg, ISSN 0933-5846 (Print) 1432-0665 (Online), Volume 46, Number 2 / February, 2007. DOI 10.1007/s00153-006-0030-2. Pages 107-121.

Received: 16 March 2006 Revised: 1 August 2006 Published online: 23 January 2007

Abstract

Let M be a nonstandard model of Peano Arithmetic with domain M and let n in M be nonstandard. We study the symmetric and alternating groups S_n and A_n of permutations of the set {0,1,...,n-1} internal to M, and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A_n and S_n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an real-valued metric on S = S_n/B_S and A = A_n/B_A (where B_S, B_A are the maximal normal subgroups of S_n and A_n identified earlier) making these groups into topological groups, and by showing that if M is aleph-1 -saturated then S and A are complete with respect to this metric.

Keywords Models of arithmetic - Nonstandard groups - Permutation groups.

Remarks

A follow-up paper is being prepared, in which many more of the properties of the group S = S_n/B_S are explored. Note that it turns out that S_n/B_S and A_n/B_A are isomorphic, an easy fact that was overlooked in the published paper.

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